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G = C89D20order 320 = 26·5

3rd semidirect product of C8 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C89D20, C4016D4, D103M4(2), C42.15D10, C8⋊C48D5, C55(C89D4), C203C83C2, (C4×D20).4C2, C10.40(C4×D4), C2.13(C4×D20), C4.77(C2×D20), (C2×D20).18C4, C20.297(C2×D4), (C2×C8).157D10, C4⋊Dic5.21C4, D101C836C2, C10.45(C8○D4), (C4×C20).13C22, C2.10(D5×M4(2)), D10⋊C4.14C4, C4.130(C4○D20), C20.246(C4○D4), (C2×C20).812C23, (C2×C40).226C22, C10.51(C2×M4(2)), C2.7(D20.2C4), (D5×C2×C8)⋊26C2, (C5×C8⋊C4)⋊7C2, (C2×C4).29(C4×D5), (C2×C8⋊D5)⋊24C2, C22.99(C2×C4×D5), (C2×C20).210(C2×C4), (C2×C4×D5).228C22, (C2×Dic5).16(C2×C4), (C22×D5).70(C2×C4), (C2×C4).754(C22×D5), (C2×C10).168(C22×C4), (C2×C52C8).303C22, SmallGroup(320,333)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C89D20
C1C5C10C20C2×C20C2×C4×D5C4×D20 — C89D20
C5C2×C10 — C89D20
C1C2×C4C8⋊C4

Generators and relations for C89D20
 G = < a,b,c | a8=b20=c2=1, bab-1=cac=a5, cbc=b-1 >

Subgroups: 446 in 124 conjugacy classes, 53 normal (47 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×2], C23 [×2], D5 [×3], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×4], M4(2) [×2], C22×C4 [×2], C2×D4, Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×5], C2×C10, C8⋊C4, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C2×M4(2), C52C8 [×2], C40 [×2], C40, C4×D5 [×4], D20 [×2], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C89D4, C8×D5 [×2], C8⋊D5 [×2], C2×C52C8 [×2], C4⋊Dic5, D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5 [×2], C2×D20, C203C8, D101C8 [×2], C5×C8⋊C4, C4×D20, D5×C2×C8, C2×C8⋊D5, C89D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, M4(2) [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×M4(2), C8○D4, C4×D5 [×2], D20 [×2], C22×D5, C89D4, C2×C4×D5, C2×D20, C4○D20, C4×D20, D5×M4(2), D20.2C4, C89D20

Smallest permutation representation of C89D20
On 160 points
Generators in S160
(1 21 75 57 85 136 112 152)(2 137 76 153 86 22 113 58)(3 23 77 59 87 138 114 154)(4 139 78 155 88 24 115 60)(5 25 79 41 89 140 116 156)(6 121 80 157 90 26 117 42)(7 27 61 43 91 122 118 158)(8 123 62 159 92 28 119 44)(9 29 63 45 93 124 120 160)(10 125 64 141 94 30 101 46)(11 31 65 47 95 126 102 142)(12 127 66 143 96 32 103 48)(13 33 67 49 97 128 104 144)(14 129 68 145 98 34 105 50)(15 35 69 51 99 130 106 146)(16 131 70 147 100 36 107 52)(17 37 71 53 81 132 108 148)(18 133 72 149 82 38 109 54)(19 39 73 55 83 134 110 150)(20 135 74 151 84 40 111 56)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 130)(22 129)(23 128)(24 127)(25 126)(26 125)(27 124)(28 123)(29 122)(30 121)(31 140)(32 139)(33 138)(34 137)(35 136)(36 135)(37 134)(38 133)(39 132)(40 131)(41 142)(42 141)(43 160)(44 159)(45 158)(46 157)(47 156)(48 155)(49 154)(50 153)(51 152)(52 151)(53 150)(54 149)(55 148)(56 147)(57 146)(58 145)(59 144)(60 143)(61 63)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(81 83)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(118 120)

G:=sub<Sym(160)| (1,21,75,57,85,136,112,152)(2,137,76,153,86,22,113,58)(3,23,77,59,87,138,114,154)(4,139,78,155,88,24,115,60)(5,25,79,41,89,140,116,156)(6,121,80,157,90,26,117,42)(7,27,61,43,91,122,118,158)(8,123,62,159,92,28,119,44)(9,29,63,45,93,124,120,160)(10,125,64,141,94,30,101,46)(11,31,65,47,95,126,102,142)(12,127,66,143,96,32,103,48)(13,33,67,49,97,128,104,144)(14,129,68,145,98,34,105,50)(15,35,69,51,99,130,106,146)(16,131,70,147,100,36,107,52)(17,37,71,53,81,132,108,148)(18,133,72,149,82,38,109,54)(19,39,73,55,83,134,110,150)(20,135,74,151,84,40,111,56), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,130)(22,129)(23,128)(24,127)(25,126)(26,125)(27,124)(28,123)(29,122)(30,121)(31,140)(32,139)(33,138)(34,137)(35,136)(36,135)(37,134)(38,133)(39,132)(40,131)(41,142)(42,141)(43,160)(44,159)(45,158)(46,157)(47,156)(48,155)(49,154)(50,153)(51,152)(52,151)(53,150)(54,149)(55,148)(56,147)(57,146)(58,145)(59,144)(60,143)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,83)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(118,120)>;

G:=Group( (1,21,75,57,85,136,112,152)(2,137,76,153,86,22,113,58)(3,23,77,59,87,138,114,154)(4,139,78,155,88,24,115,60)(5,25,79,41,89,140,116,156)(6,121,80,157,90,26,117,42)(7,27,61,43,91,122,118,158)(8,123,62,159,92,28,119,44)(9,29,63,45,93,124,120,160)(10,125,64,141,94,30,101,46)(11,31,65,47,95,126,102,142)(12,127,66,143,96,32,103,48)(13,33,67,49,97,128,104,144)(14,129,68,145,98,34,105,50)(15,35,69,51,99,130,106,146)(16,131,70,147,100,36,107,52)(17,37,71,53,81,132,108,148)(18,133,72,149,82,38,109,54)(19,39,73,55,83,134,110,150)(20,135,74,151,84,40,111,56), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,130)(22,129)(23,128)(24,127)(25,126)(26,125)(27,124)(28,123)(29,122)(30,121)(31,140)(32,139)(33,138)(34,137)(35,136)(36,135)(37,134)(38,133)(39,132)(40,131)(41,142)(42,141)(43,160)(44,159)(45,158)(46,157)(47,156)(48,155)(49,154)(50,153)(51,152)(52,151)(53,150)(54,149)(55,148)(56,147)(57,146)(58,145)(59,144)(60,143)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(81,83)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(118,120) );

G=PermutationGroup([(1,21,75,57,85,136,112,152),(2,137,76,153,86,22,113,58),(3,23,77,59,87,138,114,154),(4,139,78,155,88,24,115,60),(5,25,79,41,89,140,116,156),(6,121,80,157,90,26,117,42),(7,27,61,43,91,122,118,158),(8,123,62,159,92,28,119,44),(9,29,63,45,93,124,120,160),(10,125,64,141,94,30,101,46),(11,31,65,47,95,126,102,142),(12,127,66,143,96,32,103,48),(13,33,67,49,97,128,104,144),(14,129,68,145,98,34,105,50),(15,35,69,51,99,130,106,146),(16,131,70,147,100,36,107,52),(17,37,71,53,81,132,108,148),(18,133,72,149,82,38,109,54),(19,39,73,55,83,134,110,150),(20,135,74,151,84,40,111,56)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,130),(22,129),(23,128),(24,127),(25,126),(26,125),(27,124),(28,123),(29,122),(30,121),(31,140),(32,139),(33,138),(34,137),(35,136),(36,135),(37,134),(38,133),(39,132),(40,131),(41,142),(42,141),(43,160),(44,159),(45,158),(46,157),(47,156),(48,155),(49,154),(50,153),(51,152),(52,151),(53,150),(54,149),(55,148),(56,147),(57,146),(58,145),(59,144),(60,143),(61,63),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(81,83),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93),(101,117),(102,116),(103,115),(104,114),(105,113),(106,112),(107,111),(108,110),(118,120)])

68 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D8E8F8G8H8I8J8K8L10A···10F20A···20H20I···20P40A···40P
order12222224444444445588888888888810···1020···2020···2040···40
size1111101020111144101020222222441010101020202···22···24···44···4

68 irreducible representations

dim1111111111222222222244
type++++++++++++
imageC1C2C2C2C2C2C2C4C4C4D4D5C4○D4M4(2)D10D10C8○D4D20C4×D5C4○D20D5×M4(2)D20.2C4
kernelC89D20C203C8D101C8C5×C8⋊C4C4×D20D5×C2×C8C2×C8⋊D5C4⋊Dic5D10⋊C4C2×D20C40C8⋊C4C20D10C42C2×C8C10C8C2×C4C4C2C2
# reps1121111242222424488844

Matrix representation of C89D20 in GL4(𝔽41) generated by

32000
03200
004021
00211
,
163000
27200
00319
003438
,
1100
04000
00400
00371
G:=sub<GL(4,GF(41))| [32,0,0,0,0,32,0,0,0,0,40,21,0,0,21,1],[16,27,0,0,30,2,0,0,0,0,3,34,0,0,19,38],[1,0,0,0,1,40,0,0,0,0,40,37,0,0,0,1] >;

C89D20 in GAP, Magma, Sage, TeX

C_8\rtimes_9D_{20}
% in TeX

G:=Group("C8:9D20");
// GroupNames label

G:=SmallGroup(320,333);
// by ID

G=gap.SmallGroup(320,333);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,387,58,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=b^20=c^2=1,b*a*b^-1=c*a*c=a^5,c*b*c=b^-1>;
// generators/relations

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